Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Transfers for ramified covering maps in homology and cohomology

Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a new cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. Our transfers can also be defined in Karoubi's approach to homology and cohomology. Furthermore, we show that one can define mixed transfers from other homology or cohomology theories to the ordinary ones.

CURRENT-CURRENT CORRELATION FUNCTION ON ALGEBRAIC CURVES

We derive an explicit expression for a current-current correlation function on a Riemann surface represented as 3 sheets ramified covering over CP(1). The method used in the paper can be easily applied to more general algebraic curves. Knowledge of G(z, w) enables calculation of the expectation value of the energy momentum tensor for scalar field.